Sparse Representation and Its Applications in Blind Source Separation

In this paper, sparse representation (factorization) of a data matrix is first discussed. An overcomplete basis matrix is estimated by using the K-means method. We have proved that for the estimated overcom-plete basis matrix, the sparse solution (coefficient matrix) with minimum l1-norm is unique with probability of one, which can be obtained using a linear programming algorithm. The comparisons of the l1-norm solution and the l0-norm solution are also presented, which can be used in recoverability analysis of blind source separation (BSS). Next, we apply the sparse matrix factorization approach to BSS in the overcomplete case. Generally, if the sources are not sufficiently sparse, we perform blind separation in the time-frequency domain after preprocessing the observed data using the wavelet packets transformation. Third, an EEG experimental data analysis example is presented to illustrate the usefulness of the proposed approach and demonstrate its performance. Two almost independent components obtained by the sparse representation method are selected for phase synchronization analysis, and their periods of significant phase synchronization are found which are related to tasks. Finally, concluding remarks review the approach and state areas that require further study.